KYUNGPOOK Math. J. 2019; 59(2): 325-333
Some Generating Relations of Extended Mittag-Leﬄer Functions
Nabiullah Khan
Department of Applied Mathematics, Faculty of Engineering and Technology, Aligarh Muslim University, Aligarh-202002, India
e-mail : nukhanmath@gmail.com

Mohd Ghayasuddin
Department of Mathematics, Faculty of Science, Integral University, Lucknow226026, India
e-mail : ghayas.maths@gmail.com

Department of Applied Sciences and Humanities, Faculty of Engineering and Technology, Jamia Millia Islamia(A Central University), New Delhi-110025, India
* Corresponding Author.
Received: March 16, 2017; Revised: March 12, 2019; Accepted: March 18, 2019; Published online: June 23, 2019.

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract

Motivated by the results on generating functions investigated by H. Exton and many other authors, we derive certain (presumably) new generating functions for generalized Mittag-Leffler-type functions. Specifically, we introduce a new class of generating relations (which are partly bilateral and partly unilateral) involving the generalized Mittag-Leffler function. Also we present some special cases of our main result.

Keywords: generalized Mittag-Leﬄer’s function; hypergeometric function; generating function.
1. Introduction

In 1903, the Swedish mathematician Gosta Mittag-Leffler [7] introduced the function

$Eα(z)=∑n=0∞znΓ(αn+1),$

where z is a complex variable, Γ is the Gamma function and α ≥ 0.

The Mittag-Leffler function is a direct generalization of the exponential function to which it reduces when α = 1. For 0 < α < 1, it interpolates between the pure exponential and the hypergeometric function 1 − z−1. Its importance has been realized during the last two decades due to its involvement in problems of physics, chemistry, biology, engineering and applied sciences. The Mittag-Leffler function naturally occurs as the solution of fractional-order differential and integral equations.

Wiman [17] introduced a new generalization of Eα(z) as follows:

$Eα,β(z)=∑n=0∞znΓ(αn+β)(α,β∈ℂ,ℜ(α)>0,ℜ(β)>0),$

which is known as the Wiman function. Properties of the Wiman function Eα,β(z) and the Mittag-leffler function Eα(z) are very similar (see, [17], [1]). These Mittag-Leffler functions (which are given in (1.1) and (1.2), respectively) are entire functions of order p = 1/α. These functions play a very important role in the solution of differential equations of fractional order.

Prabhakar [9] introduced a further generalization of Eα,β(z) as follows:

$Eα,βγ(z)=∑n=0∞(γ)nΓ(αn+β)znn!,$

where α, β, γ ∈ ℂ, ℜ(α) > 0, ℜ(β) > 0, ℜ(γ) > 0, and (γ)n is the widely used Pochhammer symbol defined by

$(γ)n:=Γ (γ+n)Γ (γ)={1(n=0;γ∈ℂ{0})γ (γ+1)…(γ+n-1)(n∈ℕ;γ∈ℂ).$

For γ = 1, (1.3) is easily seen to reduce to (1.2).

The function $Eα,βγ(z)$ is the most natural generalization of the exponential function exp(z), the Mittag-Leffler function Eα(z) and Wiman’s function Eα,β(z). Kilbas and Saigo [12] and Raina [10] have investigated several properties and applications of these functions.

In a sequel to the above-mentioned works, Shukla and Prajapati [15] defined the following generalization of the Mittag-Leffler function:

$Eα,βγ,δ(z)=∑n=0∞(γ)δnΓ(αn+β)znn!(α,β,γ∈ℂ,ℜ(α)>0,ℜ(β)>0,ℜ(γ)>0, and δ∈(0,1)Uℕ),$

where $(γ)δn=Γ(γ+δn)Γ(γ)$, is the generalized Pochhammer symbol.

In 2009 and 2012, Salim [13] and Salim and Faraj [14] introduced further generalizations of the preceding functions which are given as follows:

$Eα,βγ,δ(z)=∑n=0∞(γ)nznΓ(αn+β)(δ)n(α,β,γ∈ℂ,ℜ(α)>0,ℜ(β)>0,ℜ(γ)>0, and ℜ(δ)>0)$

and

$Eα,β,pγ,δ,q(z)=∑n=0∞(γ)qnznΓ(αn+β)(δ)pn(α,β,γ∈ℂ,min{ℜ(α),ℜ(β),ℜ(γ),ℜ(δ)}>0, and q≤ ℜ(α)+p).$

Subsequently, Khan and Ahmad [6] defined the following two interesting generalizations of these functions and investigated their associated properties:

$Eα,β,δγ,q(z)=∑n=0∞(γ)qnznΓ(αn+β)(δ)n(α,β,γ∈ℂ;ℜ(α)>0,ℜ(β)>0,ℜ(γ)>0, and ℜ(δ)>0, and q∈(0,1)Uℕ)$

and

$Eα,β,ν,σ,δ,pμ,ρ,γ,q(z)=∑n=0∞(μ)ρn(γ)qnznΓ(αn+β)(ν)σn(δ)pn$

where (α, β, γ, δ, μ, ν, ρ, σ ∈ ℂ; p, q > 0; q ≤ ℜ(α) + p, and min{ℜ(α), ℜ(β), ℜ(γ), ℜ(δ), ℜ(μ), ℜ(ν), ℜ(ρ), ℜ(σ)} > 0).

2. Generating Relation

An interesting result on generating functions was given by H. Exton [3, p.147(3)]. The modified form of his result due to Pathan and Yasmeen [8] is

$exp (s+t-xts)=∑m=-∞∞∑n=m*∞smtnFnm(x),$

where

$Fnm(x)=F11(-n;m+1;x)/m!n!=Lnm(x)/(m+n)!,$

$Lnm(x)$ denotes the classical Laguerre polynomials (see [2], [12]) and

$m*=max(0,-m),(m=0,1,2,…).$

### Main result

Motivated by Exton’s result, we derive the following generating relation for the generalized Mittag-Leffler function (1.9) given as:

$Eα1,β1,ν1,σ1,δ1,p1μ1,ρ1,γ1,q1(s)Eα2,β2,ν2,σ2,δ2,p2μ2,ρ2,γ2,q2(t)Eα3,β3,ν3,σ3,δ3,p3μ3,ρ3,γ3,q3(-xts)=∑m=-∞∞∑n=m*∞smtmΓ(β1)Γ(β2)Γ(β3){ αi,βi,νi,σi,δi,piμi,ρi,γi,qiFnm(x)},$

where

${ αi,βi,νi,σi,δi,piμi,ρi,γi,qiFnm(x)}=∑k=0∞(μ1)ρ1(m+k)(γ1)q1(m+k)(μ2)ρ2(n-k)(γ2)q2(n-k)(μ3)ρ3k(γ3)q3k(β1)α1(m+k)(ν1)σ1(m+k)(δ1)p1(m+k)(β2)α2(n-k)(ν2)σ2(n-k)(δ2)p2(n-k)×(-x)k(β3)α3k(ν3)σ3k(δ3)p3k.$
Proof

On expanding the function

$M(x,s,t)=Eα1,β1,ν1,σ1,δ1,p1μ1,ρ1,γ1,q1(s)Eα2,β2,ν2,σ2,δ2,p2μ2,ρ2,γ2,q2(t)Eα3,β3,ν3,σ3,δ3,p3μ3,ρ3,γ3,q3(-xts),$

in series form, we obtain

$M(x,s,t)=∑k=0∞(μ3)ρ3k(γ3)q3k(-x)kΓ(α3k+β3)(ν3)σ3k(δ3)p3k∑i=0∞(μ1)ρ1i(γ1)q1is(i-k)Γ(α1i+β1)(ν1)σ1i(δ1)p1i×∑j=0∞(μ2)ρ2j(γ2)q2jt(j+k)Γ(α2j+β2)(ν2)σ2j(δ2)p2j.$

Replacement of ik and j +k by m and n respectively, followed by rearrangement justified by the absolute convergence of the above series, then leads to

$M(x,s,t)=∑m=-∞∞∑n=m*∞smtnΓ(β1)Γ(β2)Γ(β3)×∑k=0∞(μ1)ρ1(m+k)(γ1)q1(m+k)(μ2)ρ2(n-k)(γ2)q2(n-k)(μ3)ρ3k(γ3)q3k(β1)α1(m+k)(ν1)σ1(m+k)(δ1)p1(m+k)(β2)α2(n-k)(ν2)σ2(n-k)(δ2)p2(n-k)×(-x)k(β3)α3k(v3)σ3k(δ3)p3k,$

which is our required result.

3. Special Cases

(1) On setting μi = νi and ρi = σi (where i = 1, 2, 3) in (2.4), we get the following result for the Mittag-Leffler function defined by Salim and Faraj [14]:

$Eα1,β1,μ1,ρ1,δ1,p1μ1,ρ1,γ1,q1(s)Eα2,β2,μ2,ρ2,δ2,p2μ2,ρ2,γ2,q2(t)Eα3,β3,μ3,ρ3,δ3,p3μ3,ρ3,γ3,q3(-xts)=Eα1,β1,p1γ1,δ1,q1(s)Eα2,β2,p2γ2,δ2,q2(t)Eα3,β3,p3γ3,δ3,q3(-xts)=∑m=-∞∞∑n=m*∞smtnΓ(β1)Γ(β2)Γ(β3){ αi,βi,μi,ρi,δi,piμi,ρi,γi,qiFnm(x)},$

where

${ αi,βi,μi,ρi,δi,piμi,ρi,γi,qiFnm(x)}=∑k=0∞(γ1)q1(m+k)(γ2)q2(n-k)(γ3)q3k(-x)k(β1)α1(m+k)(δ1)p1(m+k)(β2)α2(n-k)(δ2)p2(n-k)(β3)α3k(δ3)p3k.$

(2) On setting μi = νi, ρi = σi and pi = 1 (where i = 1, 2, 3) in (2.4), we get the following interesting result:

$Eα1,β1,μ1,ρ1,δ1,1μ1,ρ1,γ1,q1(s)Eα2,β2,μ2,ρ2,δ2,1μ2,ρ2,γ2,q2(t)Eα3,β3,μ3,ρ3,δ3,1μ3,ρ3,γ3,q3(-xts)=∑m=-∞∞∑n=m*∞smtnΓ(β1)Γ(β2)Γ(β3){ αi,βi,μi,ρi,δi,1μi,ρi,γi,qiFnm(x)},$

where

${ αi,βi,μi,ρi,δi,1μi,ρi,γi,qiFnm(x)}=∑k=0∞(γ1)q1(m+k)(γ2)q2(n-k)(γ3)q3k(-x)k(β1)α1(m+k)(δ1)(m+k)(β2)α2(n-k)(δ2)(n-k)(β3)α3k(δ3)k.$

(3) On setting μi = νi, ρi = σi and qi = pi = 1 (where i = 1, 2, 3) in (2.4), we get the following known result for the Mittag-Leffler function defined by Salim in [13]:

$Eα1,β1,μ1,ρ1,δ1,1μ1,ρ1,γ1,1(s)Eα2,β2,μ2,ρ2,δ2,1μ2,ρ2,γ2,1(t)Eα3,β3,μ3,ρ3,δ3,1μ3,ρ3,γ3,1(-xts)=Eα1,β1γ1,δ1(s)Eα2,β2γ2,δ2(t)Eα3,β3γ3,δ3(-xts)=∑m=-∞∞∑n=m*∞smtnΓ(β1)Γ(β2)Γ(β3){ αi,βi,μi,ρi,δi,1μi,ρi,γi,1Fnm(x)},$

where

${ αi,βi,μi,ρi,δi,1μi,ρi,γi,1Fnm(x)}=∑k=0∞(γ1)(m+k)(γ2)(n-k)(γ3)k(-x)k(β1)α1(m+k)(δ1)(m+k)(β2)α2(n-k)(δ2)(n-k)(β3)α3k(δ3)k.$

(4) For μi = νi, ρi = σi and δi = pi = 1 (where i = 1, 2, 3), (2.4) reduces to the following known result of Kamarujjama and Khan [5] for the Mittag-Leffler function defined by Shukla and Prajapati [15]:

$Eα1,β1,μ1,ρ1,1,1μ1,ρ1,γ1,q1(s)Eα2,β2,μ2,ρ2,1,1μ2,ρ2,γ2,q2(t)Eα3,β3,μ3,ρ3,1,1μ3,ρ3,γ3,q3(-xts)=Eα1,β1γ1,q1(s)Eα2,β2γ2,q2(t)Eα3,β3γ3,q3(-xts)=∑m=-∞∞∑n=m*∞smtnΓ(β1)Γ(β2)Γ(β3){ αi,βi,μi,ρi,1,1μi,ρi,γi,qiFnm(x)},$

where

${ αi,βi,μi,ρi,1,1μi,ρi,γi,qiFnm(x)}=∑k=0∞(γ1)q1(m+k)(γ2)q2(n-k)(γ3)q3k(-x)k(1+m)k(β1)α1(m+k)(β2)α2(n-k)(β3)α3kk!.$

(5) On setting ρi = qi = σi = αi = pi = 1, μ3 = β3 and γ3 = δ3 (where i = 1, 2, 3) in (2.4), we get the following interesting result:

$E1,β1,ν1,1,δ1,1μ1,1,γ1,1(s)E1,β2,ν2,1,δ2,1μ2,1,γ2,1(t)E1,μ3,ν3,1,γ3,1μ3,1,γ3,1(-xts)=∑m=-∞∞∑n=m*∞smtn(μ1)m(γ1)m(μ2)n(γ2)nΓ(β1)Γ(β2)Γ(β3)(β1)m(ν1)m(δ1)m(β2)n(ν2)n(δ2)n×F66 [μ1+m,γ1+m,1-β2-n,1-ν2-n;1-δ2-n,1;β1+m,ν1+m,1-μ2-n,1-γ2-n;δ1+m,ν3;x],$

where pFq is the generalized hypergeometric function defined by (see, [11], [16])

$Fpq [α1,α2,…,αp;β1,β2,…,βq; z]=∑n=0∞∏j=1p(αj)n∏j=1q(βj)nznn!.$

(6) On setting ρ1 = q1 = ρ3 = q3 = α1 = σ1 = p1 = α3 = σ3 = p3 = 2, ρ2 = q2 = α2 = σ2 = p2 = 1, μ3 = β3 and γ3 = δ3 in (2.4), we get the following result:

$E2,β1,ν1,1,δ1,2μ1,2,γ1,2(s)E1,β2,ν2,1,δ2,1μ2,1,γ2,1(t)E2,μ3,ν3,2,γ3,2μ3,2,γ3,2(-xts)=∑m=-∞∞∑n=m*∞(s/4)mtn(μ12)m(μ1+12)m(γ12)m(γ1+12)m(μ2)nΓ(β1)Γ(β2)Γ(β3)(β12)m(β1+12)m(ν12)m(ν1+12)m(δ12)m(δ1+12)m(β2)n(ν2)n(δ2)n×F89 [μ1+m2,μ1+1+m2,γ1+m2,γ1+1+m2,1-β2-n,1-ν2-n;1-γ2-n,1;β1+m2,β12+1+m2,ν1+m2,ν1+1+m2,δ1+m2,δ1+1+m2,1-μ2-n,1-γ2-n,ν3;x].$

(7) On setting qi = αi = pi = 1, μi = νi, ρi = σi and γ3 = β3 (where i = 1, 2, 3) in (2.4), we get the following result:

$E1,β1,μ1,ρ1,δ1,1μ1,ρ1,γ1,1(s)E1,β2,μ2,ρ2,δ2,1μ2,ρ2,γ2,1(t)E1,γ3,μ3,ρ3,δ3,1μ3,ρ3,γ3,1(-xts)=∑m=-∞∞∑n=m*∞smtn(γ1)m(γ2)nΓ(β1)Γ(β2)Γ(β3)(β1)m(δ1)m(β2)n(δ2)n×F44 [γ1+m,1-β2-n,1-δ2-n,1;β1+m,1-γ2-n;δ1+m,δ3; x].$

(8) On setting q1 = q3 = α1 = p1 = α3 = 2, p3 = q2 = α2 = p2 = 1, γ3 = β3, μi = νi and ρi = σi (where i = 1, 2, 3) in (2.4), we get the following result:

$E2,β1,μ1,ρ1,δ1,2μ1,ρ1,γ1,2(s)E1,β2,μ2,ρ2,δ2,1μ2,ρ2,γ2,1(t)E2,γ3,μ3,ρ3,δ3,1μ3,ρ3,γ3,2(-xts)=∑m=-∞∞∑n=m*∞(s/4)mtn(γ12)m(γ1+12)m(γ2)nΓ(β1)Γ(β2)Γ(β3)(β12)m(β1+12)m(δ12)m(δ1+12)m(β2)n(δ2)n×F56 [γ1+m2,γ1+1+m2,1-β2-n,1-δ2-n,1;β1+m2,β1+1+m2,δ1+m2,δ1+1+m2,1-γ2-n,δ3; x4].$

(9) On setting q1 = q3 = α1 = α3 = 2, pi = q2 = α2 = 1, γ3 = β3, μi = νi and ρi = σi (where i = 1, 2, 3) in (2.4), we get the following result:

$E2,β1,μ1,ρ1,δ1,1μ1,ρ1,γ1,2(s)E1,β2,μ2,ρ2,δ2,1μ2,ρ2,γ2,1(t)E2,γ3,μ3,ρ3,δ3,1μ3,ρ3,γ3,2(-xts)=∑m=-∞∞∑n=m*∞(s)mtn(γ12)m(γ1+12)m(γ2)nΓ(β1)Γ(β2)Γ(β3)(β12)m(β1+12)m(δ1)m(β2)n(δ2)n×F55 [γ1+m2,γ1+1+m2,1-β2-n,1-δ2-n,1;β1+m2,β1+1+m2,1-δ2-n,δ1+m,δ3; x].$

(10) On setting α1 = 2, pi = qi = α2 = α3 = 1, γ3 = β3, μi = νi and ρi = σi (where i = 1, 2, 3) in (2.4), we get the following result:

$E2,β1,μ1,ρ1,δ1,1μ1,ρ1,γ1,1(s)E1,β2,μ2,ρ2,δ2,1μ2,ρ2,γ2,1(t)E1,γ3,μ3,ρ3,δ3,1μ3,ρ3,γ3,1(-xts)=∑m=-∞∞∑n=m*∞(s/4)mtn(γ1)m(γ2)nΓ(β1)Γ(β2)Γ(β3)(β12)m(β1+12)m(δ1)m(β2)n(δ2)n×F45 [γ1+m,1-β2-n,1-δ2-n,1;β1+m2,β1+1+m2,1-γ2-n,δ1+m,δ3; x4].$

(11) On setting ρi = qi = σi = αi = pi = μi = νi = βi = γi = δi = 1 (where i = 1, 2, 3) in (2.4), we get the following known result due to Exton [3]:

$E1,1,1,1,1,11,1,1,1(s)E1,1,1,1,1,11,1,1,1(t)E1,1,1,1,1,11,1,1,1(-xts)=exp(s+t-xts)=∑m=-∞∞∑n=m*∞smtnn!m!F11(-n;m+1;x).$
4. Concluding Remark

In our present investigation, we have studied a number of generating functions for the extended Mittag-Leffler-type functions given in [4, 6, 13, 14, 15]. The main generating function is the further generalization of the result given by Kamarujjama and Khan [5]. The results of this paper, especially (2.4), are of a general nature in the literature on generating functions.

Acknowledgements

The authors thank the constructive comments and suggestions by anonymous referees. They have contributed to improve the presentation of this manuscript. The authors wish to acknowledge R. B. Paris (Abertay University, Dundee, UK) for his assistance with the improvement of the text.

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